Determination of local contact potential difference by noncontact atomic force microscopy

ABSTRACT

A method for determining a value of a local contact potential difference by noncontact atomic force microscopy. For one or more cantilever positions above a surface of a sample: i) performing two atomic force microscopy measurements, using an oscillating cantilever, ii) thereby determining two distinct voltage values of DC voltage applied between the cantilever and the sample, and iii) obtaining a value of a local contact potential difference based, at least in part, on the two distinct voltage values determined. Wherein substantially similar distinct values indicate a substantially similar value of frequency shifts of cantilever oscillation, as measured for each of said distinct values.

CROSS REFERENCE TO RELATED APPLICATION

Pursuant to 35 U.S.C. §119, the present application claims priority toUnited Kingdom Application No. 1319250.5, filed Oct. 31, 2013, theentire contents of which are incorporated herein by reference.

FIELD OF THE INVENTION

The invention relates in general to the field of Kelvin probe forcemicroscopy (KPFM) and related microscopy techniques. In particular, itconcerns methods for determining a value of a local contact potentialdifference and for acquiring local contact potential difference maps, bynoncontact atomic force microscopy, as well as related apparatuses.

BACKGROUND OF THE INVENTION

Scanning probe microscopy is born with the invention of the scanningtunneling and the atomic force microscope. In brief, it aims at formingimages of sample surfaces using a physical probe. Scanning probemicroscopy techniques rely on scanning such a probe, e.g. a sharp tip,just above or in contact with a sample surface whilst monitoringinteraction between the probe and the surface. An image of the samplesurface can thereby be obtained. Typically, a raster scan of the sampleis carried out and the probe-surface interaction is recorded as afunction of position. Data are thus typically obtained as atwo-dimensional grid of data points. The resolution achieved varies withthe actual technique used: atomic resolution can be achieved in somecases. Typically, either piezoelectric actuators or electrostaticactuation are used to execute precise motions of the probe. Two maintypes of SPM are the scanning tunneling microscopy (STM) and the atomicforce microscopy (AFM).

In AFM techniques, forces between the tip and the surface are monitored;these are notably the short range Pauli repulsive force (incontact-mode) and the longer range attractive force (in non-contactmode, e.g., the van der Waals forces). Using AFM techniques, imaging ofthe surface topology is usually carried out in one of three modes:contact mode, non-contact mode and intermittent contact or tapping mode.In the contact mode, the probe is moved over the surface with constantcontact thus monitoring the surface by changing the height set-point. Inthe non-contact (or “noncontact”) mode, a stiff cantilever oscillateswith a small amplitude of typically less than 10 nm above the surface.Influences of the surface lead to changes in frequency and amplitude ofthe cantilever. These changes can be detected and used as the feedbacksignals. In the tapping mode, the cantilever is oscillated with a largeramplitude. Therefore also short range forces are detectable withoutsticking of the cantilever to the surface. Still, the short range forcescan be measured also in non-contact AFM, even using small amplitudes,e.g., using the so-called ‘qPlus sensor’, a tuning fork or a lengthextension resonator.

The above techniques are translated into topography by means of asensor. A common type of sensor is a bulk-component-based free-spacelaser beam deflection setup with a four quadrant photo diode acting asthe deflection sensor. Other known principles include piezoelectric,piezoresistive and thermal height sensing deflection sensors. In bothSTM and AFM, the position of the tip with respect to the surface must beaccurately controlled (e.g., to within about 0.1 Å) by moving either thesample or the tip. The tip is usually very sharp; ideally terminated bya single atom or molecule at its closest point to the surface.

Kelvin probe force microscopy (or KPFM, also known as “surface potentialmicroscopy”) and the closely related “electrostatic force microscopy”(EFM) are noncontact variants of AFM. KPFM measures work functiondifferences or the surface potential (for non-metals) between aconducting sample and a vibrating tip, at atomic scales. Note that theconcept of work function breaks down on the atomic scale; the measuredvalues correspond to the contact potential difference (“local contactpotential difference” or LCPD). The LCPD is the figure to be determinedby KPFM. It has been shown that the LCPD varies on the atomic scale. Thepotential offset between the probe tip and the surface is measured usingthe cantilever as a reference electrode that forms a capacitor with thesurface over which it is scanned, to obtain a map. Note that it does nothave to be at constant separation (and usually is not). The LCPDcorresponds to the voltage applied to the sample (with respect to thetip) that yields the minimal frequency shift Δf(V). Two methods areknown to find this minimum (i.e., the LCPD). The first method (voltagespectroscopy) consist of slowly sweeping the voltage, i.e., measuringthe Δf(V) relation and determining its minimum (voltage spectroscopy).The second method requires modulating the voltage (applying andoscillating an AC component plus a DC component) to find the DC voltagethat yields the minimum frequency shift. The second method often employslock in technique; often the second Eigen-resonance of the cantilever isused as frequency for the AC component, thus taking advantage of thecantilever's quality factor.

The electrostatic force microscopy (or EFM) directly measures the forceproduced by the electric field of the surface on a charged tip. In EFM,the frequency shift or amplitude change of the cantilever oscillation ismonitored to detect the electric field. Still, EFM and KPFM are oftenregarded as a same general noncontact variant of AFM. Both EFM and KPFMrequire the use of conductive cantilevers, typically metal-coatedsilicon or silicon nitride.

KPFM and EFM become increasingly important for the characterization ofmolecular and atomic scale electronic functional structures as theyprovide a measure of the electrostatic field and/or Work functions.Therefore, such techniques are sensitive to charges and chargedistributions. Charge resolution of about 0.1 electron charges combinedwith lateral resolution on the atomic scale was recently demonstrated in[3]. In particular, high resolution KPFM is foreseen to be of greatimportance when studying and developing future single electron logicdevices or novel materials exploiting charge transfer, e.g. materialsfor OLEDs and materials for organic solar cells. However, to obtain highresolution maps as presented in [3], measurement times are on the orderof a day for one KPFM map, corresponding to −5000 measurement points (orpixels).

The two different techniques commonly used to measure KPFM maps arecompared in [2]. Essentially, the first technique (voltage spectroscopy)provides high resolution but is slow, while the second technique (usinglock-in technique) is fast but lacks resolution.

BRIEF SUMMARY OF THE INVENTION

According to a first aspect, the present invention is embodied as amethod for determining a value of a local contact potential differenceby noncontact atomic force microscopy, the method comprising:

-   -   a. providing a sample;    -   b. for each of one or more cantilever positions above a surface        of the sample:        -   i. performing two atomic force microscopy measurements,            using an oscillating cantilever, wherein two distinct            voltage values of DC voltage applied between the cantilever            and the sample are determined, which distinct values yield a            same value of frequency shifts of cantilever oscillation, as            measured for each of said distinct values; and        -   ii. obtaining a value of a local contact potential            difference based on the two distinct voltage values            determined.

In one embodiment, determining the two distinct voltage values furthercomprises:

-   -   a. during a first AFM measurement:        -   i. selecting and storing a first voltage value of DC voltage            applied between the cantilever and the sample; and        -   ii. measuring a first frequency shift corresponding to said            first voltage value, and    -   b. during a second AFM measurement:        -   i. measuring a second frequency shift;        -   ii. adjusting a DC voltage applied between the cantilever            and the sample, to obtain a second voltage value distinct            from the first voltage value, such that the second frequency            shift measured is the same as said first measured frequency            shift; and        -   iii. storing the second voltage value.

In one embodiment, a constant value of DC voltage is used during thefirst AFM measurement, for each of said one or more positions.

In one embodiment, the method further comprises, prior to using saidconstant value of DC voltage for the first AFM measurement, selectingsaid constant value such that a corresponding frequency shift ofcantilever oscillation shall be on a same branch of the curverepresenting the frequency shift vs. an applied DC voltage, where saidbranch terminates at a principal vertex of the curve, for each of saidone or more positions.

In one embodiment, the first AFM measurement further comprises adjustinga DC voltage applied between the cantilever and the sample to obtain thefirst voltage value, the latter such that the first measured frequencyshift is the same as a predefined frequency shift.

In one embodiment, determining said two distinct voltage values of DCvoltage comprises adjusting a DC voltage during the second measurementor each of the first and second measurements, via a feedback circuitusing a frequency shift signal as a feedback signal, for determining avalue of DC voltage corresponding to a given value of frequency shift,preferably in constant height mode.

In one embodiment, the value of a local contact potential difference isobtained as an average of the two distinct voltage values stored.

In one embodiment, said one or more cantilever positions above thesurface of the sample span a subset of a volume above the surface, saidsubset being preferably one of: a 3D, a 2D or a 1D portion of saidvolume.

According to a second aspect, the invention is embodied as a method ofacquiring a frequency shift map, comprising all the steps of any one ofthe above embodiments of the first aspect, and further comprising:performing a third AFM measurement for each of said one or morecantilever positions above the surface of the sample, using the localcontact potential difference as determined for said each of severalcantilever positions, and wherein, preferably, the first AFM measurementis performed for each of several cantilever positions above the surfaceof the sample, and then the second AFM measurement is performed for eachof the same several cantilever positions, and then the third AFMmeasurement is performed for each of the same several cantileverpositions.

According to a third aspect, the invention is embodied as an atomicforce microscopy apparatus, configured for performing atomic forcemicroscopy measurements using an oscillating cantilever, the apparatuscomprising:

-   -   a. a frequency shift measurement unit configured for measuring a        frequency shift of cantilever oscillation;    -   b. a circuit connected to the frequency shift measurement unit,        and configured to:        -   i. determine a DC voltage to be applied by the apparatus            between a cantilever and a sample for an AFM measurement;            and        -   ii. adjust such a DC voltage such that a value of frequency            shift as measured by the frequency shift measurement unit            matches a reference frequency shift; and    -   c. memory means storing one or more pairs of distinct values of        DC voltages, corresponding to a same value of frequency shift of        cantilever oscillation;    -   d. processing means, connected to the memory means to access        said one or more pairs of voltage values, and configured for        determining values of local contact potential differences based        on said one or more pairs.

In one embodiment, the circuit is configured as a feedback circuit,connected to the frequency shift measurement unit to use a frequencyshift signal as a feedback signal and adjust a DC voltage such that avalue of frequency shift measured by the frequency shift measurementunit is the same as a reference frequency shift.

In one embodiment, the circuit has two selectable modes, wherein:

-   -   a. in a first one of the selectable modes, the circuit does not        adjust the DC voltage to be applied; and    -   b. in a second one of the selectable modes, the circuit adjusts        the DC voltage to be applied, such that a value of frequency        shift measured by the frequency shift measurement unit matches a        reference frequency shift.

In one embodiment, the processing means are further configured fordetermining a value of the local contact potential difference as anaverage of two distinct DC voltage values in a given one of said one ormore pairs.

In one embodiment, the apparatus is a Kelvin probe force microscopyapparatus or an electrostatic force microscopy apparatus.

In one embodiment, the apparatus is further configured for acquiring afrequency shift map by performing AFM measurement (more preferablyconstant height AFM measurements) using local contact potentialdifferences as determined by the processing means.

BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a view of a simplified representation of selected componentsof an atomic force microscopy apparatus, as used in embodiments;

FIG. 2 is a diagram schematically illustrating selected components of anatomic force microscopy apparatus according to embodiments;

FIG. 3 is a flowchart illustrating high-level steps of a method fordetermining a value of a local contact potential difference, accordingto a first class of embodiments;

FIG. 4 is a variant to FIG. 3, illustrating high-level steps of a methodaccording to a second class of embodiments;

FIG. 5 shows a parabola representing the frequency shift Δf(V) as afunction of a DC voltage applied between the cantilever and the sample,and illustrates two distinct voltage values that yield a same value offrequency shift of cantilever oscillation; and

FIG. 6 shows two parabolas Δf(V) measured at two different positions aand b, as well as various values to be determined according to presentmethods.

DETAILED DESCRIPTION OF THE INVENTION

Apparatuses and methods embodying the present invention will now bedescribed, by way of non-limiting examples, and in reference to theaccompanying drawings. Technical features depicted in the drawings arenot to scale.

The following description is structured as follows. First, generalembodiments and high-level variants are described (sect. 1). The nextsection addresses more specific embodiments and technical implementationdetails (sect. 2).

Sect. 1. General Embodiments and High-Level Variants

As evoked in introduction, two different methods are commonly used sofar to measure KPFM maps. One of these methods (using lock-intechniques) is fast but lacks resolution, while the other one (voltagespectroscopy) provides high resolution but is slow. The novel methodsproposed here add to the high resolution of the latter a fastermeasurement capability.

First, a brief review the advantages and drawbacks of the existing KPFMmethods. KPFM is a scanning probe method related to non-contact atomicforce microscopy (hereafter “NC-AFM”). In NC-AFM, a cantilever isoscillated at its resonance frequency. The cantilever holds the tip(probe). When the tip is approached to the surface the resonance ingeneral will be changed due to the interaction between sample and tip.This frequency shift Δf is the measurement signal in NC-AFM. In general,Δf depends on the position (x, y, z) above the sample and the biasvoltage V applied to the sample with respect to the tip: Δf=Δf (V, x, y,z) or Δf=Δf(V) for short. In KPFM, the relationship between Δf and V isessential and one wants to measure two properties: (i) The voltage V*that yields the smallest absolute value of the frequency shift as afunction of V and (ii) the frequency shift at this voltage: Δf*=Δf(V*).This voltage V* is also called “local contact potential difference” (orLCPD). The LCPD is closely related to the so-called nulling potential.However, the electrostatic forces are not necessarily nullified in KPFM,they are only minimized (they are effectively nullified only when aplate capacitor is used).

As it can be realized, Δf(V) is essentially symmetric near its peak (orvertex); Δf(V) has the form of a parabola. The peak of the parabola (V*,Δf*) corresponds to the values sought (see FIGS. 5-6); the peak variesas a function of the tip position (x, y, z).

For example, FIG. 6 shows a schematic of two Δf(V) parabolas, asobtained for two different positions (a and b). The different values tobe determined are indicated: (V*(a), Δf*(a)=Δf(V*(a))) and (V*(b),Δf*(b)=Δf(V*(b))). In the case of a KPFM map acquired in thespectroscopic mode, the curve Δf(V) is measured for each pixel of themap and the values (V*, Δf*) are determined pixel by pixel [2,3].

Next, high resolution KPFM assumes that some conditions are met [3].Most importantly, these are:

1) A high stability, usually provided by working at cryogenictemperatures. Moreover, KPFM maps are acquired in constant height mode,which requires small thermal drift. Thus, there is no feedback circuitneeded to control the tip height. Additionally, interpretation of imagesis usually facilitated by measurements at constant height; and

2) To increase resolution and suppress noise in the frequency shift (Δf)channel, the Δf-channel should be measured with low bandwidth. For thisreason KPFM measurements using lock-in techniques, modulating the biasto track the LCPD during movement of the tip, do not yield a resolutioncomparable to the spectroscopic method. Another reason for thecomparably smaller resolution of this mode is that the DC voltage isalways at the LCPD, but in this region the slope of Δf(V) is minimal,therefore the error in the determination of the minimum of |Δf(V)| islarge.

The novel methods proposed here allow the acquisition speed of LCPD mapsto be increased with respect to the spectroscopic mode, with unimpairedhigh resolution. These methods basically exploit the observed symmetryof the parabola Δf(V) near its peak. The underlying idea is to performtwo atomic force microscopy measurements at the same position (x, y, z)to determine two distinct voltage values V1, V2 on each side of thepeak. This is done for each cantilever position (x, y, z) of interest.To obtain a 2D map, the (x, y, z) positions preferably span a plane. Thetwo distinct voltage values are determined such as to yield a same valueof frequency shift of the cantilever oscillation. The local contactpotential difference V* is then easily obtained from the two distinctvoltage values V1, V2, e.g., as an average thereof. The frequency shiftcan later be acquired at the same position, and for the voltage valueV*, which drastically reduce the time of acquisition.

In reference to FIG. 1-6, a first aspect of the invention is nowdescribed in detail; this aspect concerns a method for determining avalue of a local contact potential difference by noncontact atomic forcemicroscopy.

First, a sample 30 is provided, step S10, the surface 32 of which is tobe analyzed. Then, a series of measurements shall be performed for eachcantilever position (x, y, z) above the surface 32 of the sample (stepS20). In general these can be any positions above the sample surface 32.To obtain a 2D map one will preferably use the constant height mode,i.e. move the tip on a plane parallel to the surface 32 of the sample.In variants, and in particular if the surface 32 exhibits largecorrugations, one may move the tip on a surface that essentially followsthe topography of the sample surface 32.

In addition, measurements could also be performed at different tipheights (different z), in order to obtain 3D datasets. Thus, the AFMmeasurements need not be restricted to a given plane or surface (thelatter should not be confused with the surface 32 of the sample).However, the following description mostly assumes constant heightmeasurements, i.e., measurement points on a plane parallel to the samplesurface (as this will generally be the case), for the sake ofillustration but without prejudice.

For instance, two constant height AFM measurements steps S31, S32) areperformed using an oscillating cantilever 10, for each cantileverposition (x, y, z) of interest above the surface 32 of the sample. Saidconstant height measurements allow two distinct voltage values V1, V2 ofDC voltage (applied between the cantilever and the sample) to bedetermined, steps S311-S322. Said two values are determined such as tobe on each side of the peak of the Δf(V) curve. To ensure this, oneneeds to determine two values V1, V2, which yield a same value offrequency shift Δf(V1)=Δf(V2). As evoked above, these two measurementscould, in variants, follow the topography of the sample surface 32. Asillustrated in FIG. 1, the DC voltages are measured across junction 20with voltage connector 20 a being attached to cantilever 10/tip 12 andvoltage connector 20 b being attached to sample 30.

The frequency shift is the difference of the measured frequency (nearthe surface 32, with a voltage applied) with respect to the frequency ofthe free (unperturbed) sensor (or in principle with respect to anyreference frequency, the absolute value of the base frequency beingunimportant). The measurement signal is a frequency difference(therefore called shift). The shift varies when moving the cantileverposition or applying a voltage.

As the skilled person may appreciate, said two values V1, V2 need besufficiently distinct, to ensure that they are located on opposite sidesof the peak. Namely, the minimal shift between V1 and V2 needs besignificantly larger (e.g., at least a factor 10) than the availablevoltage resolution (which notably depends on the resolution for Δf). Thetwo values V1, V2 may therefore be subject to a minimal, predeterminedshift between them. This predetermined shift can otherwise be determinedby trial-and-error.

Next, step S40, the value of local contact potential difference (or LCPDvalue) V* can be determined based on the values V1, V2 previouslyobtained. For example, if the Δf(V) curve is assumed to be perfectlysymmetric about the peak (at least in the range of voltage valuesconsidered), then the LCPD value is a simple arithmetic averageV*=(V1+V2)/2. If the Δf(V) curve is known to exhibit some asymmetryabout the peak, a predetermined model could be used, like V*=(rV1+(1−r)V2)/2, where rε]0, 1[. More elaborated schemes could becontemplated, depending on the exact nature of the curve. A similar,modified scheme may for instance also be used to correct a known bias inthe measurement of V1 and V2, etc.

The present approach is time efficient, because two distinct voltagevalues suffice, in principle, to determine the LCPD value V*. Inparticular, the methods proposed herein makes it possible to markedlyincrease the acquisition speed of LCPD maps with respect to thespectroscopic mode because the bias does not need to be ramped throughthe whole bias range at every pixel. Meanwhile, present methodsessentially preserve the resolution. Incidentally, they preserve the tip12, especially when tip functionalization is needed, e.g., to yield veryhigh resolution. This is due to the limited number of measurementsneeded here, compared with spectroscopic methods, where the bias voltageis ramped quickly and often, such that the tip can suffer damage (e.g.,the front atom or molecule gets lost).

Note that one could, in principle, achieve the same by (i) measuringthree (or more) frequency shift values (anywhere on the parabola),fitting a parabola and extrapolating the vertex coordinates, or still by(ii) measuring a first frequency shift and then its first and secondderivatives at the same point. However, as present inventors haverealized, such alternatives would be much more consequently impacted bynoise effects than the proposed approach, which benefits from fortunatecompensations of errors, owing to the symmetry of the measurements aboutthe principal vertex of the Δf(V) curve.

Present methods can typically be applied to Kelvin probe forcemicroscopy (also known as surface potential microscopy) or electrostaticforce microscopy (EFM).

In the following, two main classes of embodiments will be described. Inthe first class of embodiments, the first constant height measurementprovides a frequency shift Δf1 that shall serve as a reference for thesecond measurement. In the second class of embodiments, both the firstand second measurements use the same, predetermined frequency shiftvalue Δf0 as a reference.

The first class of embodiments is now described in detail, in referenceto FIG. 3. The determination of the two distinct voltage values V1, V2requires two AFM measurements S31, S32, e.g., constant heightmeasurements. During the first measurement S31, a first voltage V1 isselected and stored, S311, and a first frequency shift Δf1 is measured,S312. During the second measurement S32: a second frequency shift Δf2 ismeasured S321; then the DC voltage is adjusted S322, in order to obtainthe second voltage value V2, distinct from V1, such that Δf2 as measuredat step S321 is the same as Δf1. One may for instance aim at aresolution on the order of 10 mV for V. Still, there is no need to set athreshold in the measurement. Trying to obtain Δf1=Δf2 can be subject toa relevant tolerance t, as the skilled person may appreciate, that is,|Δf1−Δf2|<t, where t can be estimated from the resolution sought for V(e.g., 10 mV). Finally, the second voltage value V2 is stored, stepS323.

This class of embodiments is advantageous inasmuch as no adjustment isneeded during the first measurement, which increases the process speed.

Preferably, a constant value of DC voltage is used during the firstconstant height measurement S31, for each of said one or more positions.There is no strict need to change the voltage while scanning an area onthe surface 32. The voltage does not need to be modulated in this mode.This allows the usage of a small bandwidth of the phase-locked loop (orPLL) for the determination of the frequency shift. Accordingly, thenoise level in the frequency shift can be reduced and the resolution inthe determination of V* and Δf* is increased.

More preferably, said constant value of DC voltage is selected such asto ensure that the first frequency shift stay on a same branch of theΔf(V) curve for each position (x, y). I.e., the first frequency shiftshall remain on a same side of the principal vertex of the curve. Theadjusting mechanism (e.g., a feedback circuit) then knows what voltagedirection is to be explored to find the second voltage. To illustratethis, consider FIG. 5: if V1 is known to be most certainly located onthe left-hand side of the peak, then the adjusting circuit knows itneeds to increase the voltage to locate the symmetric voltage value V2that gives rise to the same value of frequency shift Δf(V2)=Δf(V1). Invariants, if it is not known on which side of the parabola the firstvoltage value V1 a priori is, it is only needed to calculate the slopeat V1 to determine which direction to explore to find the symmetricvalue V2.

Referring now to FIG. 4, in the second class of embodiments, also thefirst constant height measurement S31 comprises adjusting S312 the DCvoltage applied between the cantilever and the sample to obtain V1,where V1 is such that Δf1=Δf(V1)=Δf0, and Δf0 is a predefined frequencyshift. Accordingly, the Δf setpoint does not need to be varied duringthe second measurement. Then, during the second constant heightmeasurement S32, the second frequency shift Δf2 is measured S321 and theDC voltage adjusted S322, to obtain a value V2 distinct from V1, suchthat Δf2 (as measured S321) is the same as Δf0 (and effectively the sameas Δf1 since, ideally, Δf2=Δf1=Δf0). In practice, one can use the sameΔf setpoint for both measurements (instead of using the value measuredduring the first measurement as a setpoint for the second measurement).In other words, Δf0 can be used as a reference value for bothmeasurements. The first and second constant height measurement steps areessentially identical in that case, which results in reducingmeasurement artifacts. Still, a drawback of this variant is that it issomewhat slower than the previous class (but at most by a factor of 2),because now voltage adjustment is needed at each of the two constantheight measurements.

Variants are now described which concern both classes of embodiments.For instance, referring to FIG. 2, the determination of a voltage valuemay most practically use a feedback circuit 26, wherein the frequencyshift is used as a feedback signal to determine the second (or each ofthe first and second) voltage values. For example, in FIG. 3, feedingS313 feeds the voltage to feedback circuit 26 with the first frequencyshift Δf1 value. In addition, the cantilever positions explored abovethe surface 32 may span a volume or a subset thereof (e.g., a 1D, 2D or3D subset, since measurements might also be performed at differentheights). Said subset may for instance be a 2D area, a line (involvingseveral single-point measurements at different positions of thecantilever, which are distributed along a single line) or a pixelthereof. Single-point measurements (pixel) are less time efficient.

Referring now to FIG. 3-5, the above methods may be used as a preambleto methods for acquiring Frequency shift maps, according to a secondaspect of the invention. To that aim, and in addition to the stepsdescribed above, additional steps consist in performing a third constantheight AFM measurement S33 for each cantilever position of interest. TheLCPD value V* as determined earlier is used to measure Δf(V*) for eachcantilever position S50.

Note that the embodiments of FIGS. 3, 4 reflect schemes where a givenposition of the tip is selected and the three AFM measurements areperformed for that position, before moving to the next position.However, one may prefer to do a “first AFM measurement” (S31) at each ofseveral positions and then a “second AFM measurement” (S32) at thosepositions, etc. The corresponding flowcharts are not depicted, forsimplicity (they would require several additional arrows). Therefore,and in variants to the algorithms of FIGS. 3, 4, a sequence of firstmeasurements may be performed first, to determine V1 for each of severalcantilever positions, before performing a sequence of secondmeasurements (to determine V2 for the same positions). Then, the thirdconstant height measurement is performed to measure Δf(V*) for each ofthe same positions (S33). Each of the first, second and thirdmeasurements can be performed for a whole volume (above the surface 32)or a (1D, 2D or 3D) subsection thereof, e.g., to obtain a whole map.Doing it pointwise is more time consuming, though not excluded. Atrade-off may be found, e.g., each measurement could be done stepwise,e.g., line by line. Smaller steps take more time, but are advantageousif there is a drift, since decreasing the time between two measurementsat the different voltages decreases the effect of drift.

Note that the drift can not only be a thermal drift but also be due topiezo creep or piezo hysteresis. In general, drift decreases withdecreasing temperatures. Therefore working at cryogenic temperatures isbeneficial and allows for larger steps to be used, i.e., performing“first AFM measurement” at several positions before performing “secondAFM measurements” at the same positions, etc.

Referring back to FIGS. 1, 2, another aspect of the invention isdescribed which concerns an AFM apparatus 100. The latter is generallyconfigured for performing AFM measurements using an oscillatingcantilever. As noted earlier and assumed henceforth, the apparatus 100is preferably designed for enabling constant height AFM measurements. Itnotably comprises a frequency shift measurement unit 21-23, generallyconfigured for measuring a frequency shift of cantilever oscillation.This unit may notably comprise a deflection sensor 21, a bandpass filter22 and a Δf demodulator (PLL) 23, arranged as in FIG. 2, whichcomponents are known per se.

In order to enable methods such as described above, the apparatusfurther comprises a circuit 26, suitably connected to the frequencyshift measurement unit and notably configured to:

Determine a DC voltage to be applied by the apparatus (between thecantilever and the sample), when performing a constant heightmeasurement such as described earlier; and

Adjust such a DC voltage such that a value of frequency shift Δf2 asmeasured by the frequency shift measurement unit during a constantheight measurement matches a reference frequency shift (be it apredetermined value Δf0 or a value corresponding to a previouslymeasured shift Δf1).

The apparatus may further comprise an automatic gain control unit 24 anda phase shifter 25, as usual in the art and as illustrated in FIG. 2.Frequency modulation mode is used: the amplitude can be kept constantusing a separate feedback loop.

Furthermore, the apparatus typically comprises memory means (not shown),for storing one or more pairs of distinct values of DC voltages(corresponding to a same value of frequency shift measured). Processingmeans are provided too (not shown), which cooperate with the memorymeans to access said pairs of voltage values. The processing means arenotably configured for determining values of LCPD values V* based onsuch pairs, as explained earlier. The memory means and processing meanscan be integrated. The circuit 26 itself may be at least partlyintegrated with the processing means or use the processing capability ofthe processing means to perform its function, or part thereof.

Various algorithms may be used by the circuit 26 to adjust and determinea voltage value V(Δf), i.e., corresponding to a given frequency shiftvalue. For instance, sampling techniques and extrapolation techniquesmay be involved. Most advantageous, however, is to have the circuit 26configured as a feedback circuit, as evoked earlier. The feedbackcircuit 26 can be suitably connected to the frequency shift measurementunit 21-23, as seen in FIG. 2, such as to use the frequency shift as afeedback signal to determine the second (or each of the first andsecond) voltage values. The circuit 26 can thus adjust a DC voltage suchthat a value of frequency shift Δf1, Δf2 measured by the frequency shiftmeasurement unit be the same as a reference frequency shift Δf0, Δf1.

Note that, with respect to a usual frequency modulation AFM, only thefeedback circuit needs be modified to implement the essential, novelfunctions described herein. Namely, a “voltage” feedback (using Δf asfeedback signal) is used instead (or in addition to) a usual “z feedbackcircuit”, as known in the constant frequency mode in usual frequencymodulation AFM. The property monitored is accordingly a voltage insteadof the topography.

In embodiments, the circuit 26 has two selectable modes, wherein:

In the first mode, the circuit does not adjust the DC voltage to beapplied; and

In the second mode, the circuit adjusts the DC voltage to be applied,such that a value of frequency shift Δf2 (as measured by the frequencyshift measurement unit) matches the reference frequency shift (be it Δf0or Δf1).

The ability to pass from one mode to the other notably allows forimplementing the first class of embodiments described earlier.

The apparatus is otherwise configured for acquiring a Frequency shiftmap: to that aim, constant height AFM measurement are performed usingLCPD values as determined by the processing means.

The above embodiments have been succinctly described in reference to theaccompanying drawings and may accommodate a number of variants. Severalcombinations of the above features may be contemplated. Examples aregiven in the next section.

2. Specific Embodiments/Technical Implementation Details

To illustrate the above concepts, an example of acquisition of a LCPDmap and a Δf* map is now described in reference to FIGS. 5 and 6.

During a sequence of first constant height measurements, a reference Δfmap is measured at constant bias V1 (FIG. 5). The bias V1 is chosen insuch a way that for all cantilever positions of the map one stays on thesame flank of the Δf(V) parabola. As evoked in the previous section, V1can be chosen to always stay on the rising flank (∂[Δf(V)]/∂V>0) of theparabolas. Thus, the complete first image is measured at the same biasvoltage V1, i.e. like a usual NC-AFM constant height measurement.

During a sequence of second constant height measurements, the second mapis acquired for the same positions as already used for the firstsequence, but at different voltages V2. Note that the coordinates x, y,and z are the same at both measurements. In general the positions can beany positions above the sample surface 32. To obtain a 2D map of thesample surface 32, one will preferably use the constant height mode,i.e. move the tip on a predetermined plane that is parallel to thesurface 32. Or, for samples with large corrugations, the tip may bemoved on a surface that essentially follows the topography of the samplesurface 32, as noted earlier. Still, measurements along 3D paths(realizing, e.g., 3D maps, vertical lines, or tilted planes) can also beperformed. The other, distinct values V2 are on the other flank of theΔf(V) parabola compared to the first measurements. In the secondmeasurements, the feedback circuit 26 is used to determine and apply thevoltage V2 that yields the same Δf as V1, when applied at a givenposition. In other words, the feedback circuit finds and applies V2 suchthat Δf(V2)=Δf(V1), i.e., for each point (x1, y1, z1), one hasΔf(V2)|_((x1,y1,z1))=Δf(V1)|_((x1,y1,z1)). The LCPD map is obtainedafter these second measurements by the values V*=(V1+V2)/2 (for eachposition), i.e., the set of values V*(x, y, z) is the LPCD map.

After this second sequence, a third sequence of constant heightmeasurements is performed to acquire the Δf* map, i.e., Δf* is measuredfor each LCPD value V* at each position.

FIG. 6 considers two positions a and b as examples. In the first step,the voltage is set to the value V1 for each position (a and b) and thevalues Δf(V1, a) and Δf(V1, b) are measured. In the second step, afeedback circuit on V(Δf) finds the voltages V2 such that Δf(V2,a)=Δf(V1, a) and Δf(V2, b)=Δf(V1, b) with V1<V2 if ∂[Δf(V)]/∂V|_(V1)>0and V1>V2 if ∂[Δf(V)]/∂V|_(V1)<0. In variants, the feedback circuit maybe used to measure the frequency shift at each position a and b suchthat Δf(V2, a)=Δf(V1, a)=Δf0 and Δf(V2, b)=Δf(V1, b)=Δf0. Thepredetermined value Δf0 does, strictly speaking, not need to be the sameat each position a and b.

While the present invention has been described with reference to alimited number of embodiments, variants and the accompanying drawings,it will be understood by those skilled in the art that various changesmay be made and equivalents may be substituted without departing fromthe scope of the present invention. In particular, a feature(device-like or method-like) recited in a given embodiment, variant orshown in a drawing may be combined with or replace another feature inanother embodiment, variant or drawing, without departing from the scopeof the present invention. Various combinations of the features describedin respect of any of the above embodiments or variants may accordinglybe contemplated, that remain within the scope of the appended claims. Inaddition, many minor modifications may be made to adapt a particularsituation to the teachings of the present invention without departingfrom its scope. Therefore, it is intended that the present invention notbe limited to the particular embodiments disclosed, but that the presentinvention will include all embodiments falling within the scope of theappended claims. In addition, many other variants than explicitlytouched above can be contemplated. For example, the predeterminedreference frequency shift value Δf0 may vary, e.g., be optimized as afunction of the position (x, y, z).

The invention claimed is:
 1. A method for determining a value of a localcontact potential difference by noncontact atomic force microscopy, themethod comprising: for one or more cantilever positions above a surfaceof a sample: performing two atomic force microscopy measurements, usingan oscillating cantilever, thereby determining two distinct voltagevalues of DC voltage applied between the cantilever and the sample,wherein substantially similar distinct values indicate substantiallysimilar values of frequency shifts of cantilever oscillation, asmeasured for each of said distinct values; and obtaining a value of alocal contact potential difference based, at least in part, on the twodistinct voltage values that were determined.
 2. The method of claim 1,wherein determining the two distinct voltage values further comprises:during a first AFM measurement: selecting and storing a first voltagevalue of DC voltage applied between the cantilever and the sample;measuring a first frequency shift corresponding to said first voltagevalue; and storing a corresponding first voltage value of DC voltageapplied between the cantilever and the sample; and during a second AFMmeasurement: measuring a second frequency shift; adjusting a DC voltageapplied between the cantilever and the sample, to obtain a secondvoltage value distinct from the first voltage value, such that thesecond frequency shift measured is the same as said first measuredfrequency shift; and storing the second voltage value.
 3. The method ofclaim 2, wherein a constant value of DC voltage is used during the firstAFM measurement, for each of said one or more positions.
 4. The methodof claim 3, further comprising, prior to using said constant value of DCvoltage for the first AFM measurement, selecting said constant valuesuch that a corresponding frequency shift of cantilever oscillationshall be on a same branch of the curve representing the frequency shiftvs. an applied DC voltage, where said branch terminates at a principalvertex of the curve, for each of said one or more positions.
 5. Themethod of claim 2, wherein the first AFM measurement further comprisesadjusting a DC voltage applied between the cantilever and the sample toobtain the first voltage value, the latter such that the first measuredfrequency shift is the same as a predefined frequency shift.
 6. Themethod of claim 1, wherein determining said two distinct voltage valuesof DC voltage comprises adjusting a DC voltage during the secondmeasurement or each of the first and second measurements, via a feedbackcircuit using a frequency shift signal as a feedback signal, fordetermining a value of DC voltage corresponding to a given value offrequency shift, preferably in constant height mode.
 7. The method ofclaim 1, wherein the value of a local contact potential difference isobtained as an average of the two distinct voltage values stored.
 8. Themethod of claim 1, wherein said one or more cantilever positions abovethe surface of the sample span a subset of a volume above the samplesurface, said subset being preferably one of: a 3D, a 2D, or a 1Dportion of said volume.
 9. The method of claim 1, the method furthercomprising: acquiring a frequency shift map by performing a third AFMmeasurement for each of said one or more cantilever positions above thesurface of the sample, using the local contact potential difference asdetermined for said each of several cantilever positions, wherein, thefirst AFM measurement is performed for each of several cantileverpositions above the surface of the sample, and then the second AFMmeasurement is performed for each of the same several cantileverpositions, and then the third AFM measurement is performed for each ofthe same several cantilever positions.
 10. An atomic force microscopyapparatus, configured for performing atomic force microscopymeasurements using an oscillating cantilever, the apparatus comprising:a frequency shift measurement unit configured for measuring a frequencyshift of cantilever oscillation; a circuit connected to the frequencyshift measurement unit, the circuit being configured to: determine a DCvoltage to be applied by the apparatus between a cantilever and a samplefor an AFM measurement; and adjust such a DC voltage such that a valueof a frequency shift, as measured by the frequency shift measurementunit, matches a reference frequency shift; a memory that is configuredto store one or more pairs of distinct values of DC voltages thatcorrespond to a same value of frequency shift of cantilever oscillation;and a processor connected to the memory to access said one or more pairsof voltage values, the processor being configured to determine values oflocal contact potential differences based, at least in part, on said oneor more pairs.
 11. The apparatus of claim 10, wherein the circuit is i)configured as a feedback circuit; ii) connected to the frequency shiftmeasurement unit; iii) configured to use a frequency shift signal as afeedback signal; and iv) configured to adjust a DC voltage such that avalue of frequency shift measured by the frequency shift measurementunit is the same as a reference frequency shift.
 12. The apparatus ofclaim 10, wherein the circuit has two selectable modes, wherein: in afirst one of the selectable modes, the circuit does not adjust the DCvoltage to be applied; and in a second one of the selectable modes, thecircuit adjusts the DC voltage to be applied, such that a value offrequency shift measured by the frequency shift measurement unit matchesa reference frequency shift.
 13. The apparatus of claim 10, wherein theprocessor is further configured to determine a value of the localcontact potential difference as an average of two distinct DC voltagevalues in a given one of said one or more pairs.
 14. The apparatus ofclaim 10, wherein the apparatus is at least one of a Kelvin probe forcemicroscopy apparatus or an electrostatic force microscopy apparatus. 15.The apparatus of claim 10, wherein the apparatus is further configuredto acquire a frequency shift map by performing one or both of AFMmeasurement and constant height AFM measurements, using local contactpotential differences as determined by the processing means.